| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 2·13-s + 2·15-s + 21-s − 25-s − 27-s + 2·29-s + 8·31-s + 2·35-s − 6·37-s + 2·39-s + 6·41-s + 4·43-s − 2·45-s + 49-s − 14·53-s − 8·59-s + 14·61-s − 63-s + 4·65-s + 4·67-s + 8·71-s − 10·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s − 1.92·53-s − 1.04·59-s + 1.79·61-s − 0.125·63-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2257720898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2257720898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.48025565085420, −11.93461238729572, −11.67199889419207, −11.10306208676709, −10.80093376226092, −10.17443570366075, −9.831056053287646, −9.395260134873124, −8.808184036032298, −8.274726494670563, −7.811510976551087, −7.503854882250906, −6.872300449037361, −6.486017547289278, −6.086078989712077, −5.394323177635336, −4.970652533684839, −4.498785327612005, −3.899046878669626, −3.641295480028318, −2.662124961452605, −2.607441827339703, −1.519375565888635, −1.014959236955846, −0.1443692592918506,
0.1443692592918506, 1.014959236955846, 1.519375565888635, 2.607441827339703, 2.662124961452605, 3.641295480028318, 3.899046878669626, 4.498785327612005, 4.970652533684839, 5.394323177635336, 6.086078989712077, 6.486017547289278, 6.872300449037361, 7.503854882250906, 7.811510976551087, 8.274726494670563, 8.808184036032298, 9.395260134873124, 9.831056053287646, 10.17443570366075, 10.80093376226092, 11.10306208676709, 11.67199889419207, 11.93461238729572, 12.48025565085420
